Problem: s involving definite integrals (algebraic) AP.CALC: CHA‑4 (EU), CHA‑4.D (LO), CHA‑4.D.1 (EK), CHA‑4.D.2 (EK), CHA‑4.E (LO), CHA‑4.E.1 (EK) Google Classroom Facebook Twitter Email Problem A cart is slowing down at a rate of $2t+60$ centimeters per second per second (where $t$ is the time in seconds). By how many centimeters per second does the cart slow down between $t=5$ and $t=15$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $20$ (Choice B) B $75$ (Choice C) C $800$ (Choice D) D $875$
Answer: Letting $z(t)$ be the decrease in velocity of the cart at second $t$ from the velocity at $t=0$, we are given that $z'(t)=2t+60$. We want to find $z(15)-z(5)$. According to the Fundamental Theorem of Calculus, $\begin{aligned} z(15)-z(5)&=\int_{5}^{15} z'(t)dt \\\\ &=\int_{5}^{15}(2t+60)dt \end{aligned}$ $\int_{5}^{15}(2t+60)dt=800$ In conclusion, between $t=5$ and $t=15$ the cart slows down by $800$ centimeters per second.